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Definition

Symbol-free definition

Given a solvable group, we define its derived length or solvable length as follows:

It is the length of the derived series of the group. Note here that by length of the series, we mean the number of successive inclusions, so the length is one less than the actual number of subgroups in the derived series.

It is the minimum possible length of a subnormal series from the trivial subgroup to the whole group such that all the quotients in the series are abelian groups.

When we say that a group has derived length , we typically mean that it has solvable length at most .